The Change of the Vibration Entropy Due to the Disordering of an Ordered Simple-Cubic Alloy
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THE CHANGE OF THE VIBRATION ENTROPY DUE TO THE DISORDERING OF AN ORDERED SIMPLE-CUBIC ALLOY
A.A.H.J. Waegmaekers and H. Bakker Natuurkundig Laboratorium der Universiteit van Amsterdam, 1018 XE Amsterdam, The Netherlands
Valckenierstraat
65,
ABSTRACT Within a model of central and noncentral vibrational entropy,
AS,
is
calculated,
forces the exact change of the
both as function of the degree of
order and as a function of the diverse force constants. A linear relationship between AS and the short-range order parameter is
found.
Results of AS as a
function of the force constants are compared with AS as calculated by an empirical formula based on a modified Einstein crystal model. INTRODUCTION On the basis of statistical mechanics several analytical approximations have been developed to describe order-disorder transformations. formation will,
fect the lattice vibrations. and this is is
Such a trans-
apart from the configurational entropy and energy,
also af-
The spectrum of lattice vibrations will change,
accompanied by a change in the vibrational entropy.
However,
it
often assumed intuitively that this contribution to the total change of
the entropy is In
negligible.
the past two attempts have been undertaken to make an estimate of the
influence of
lattice vibrations on thermodynamic properties at an order-dis-
order transformation.
Booth and Rowlinson
model of uncoupled vibrations,
[i] have shown,
using the Einstein
that the size of the discontinuity in the spe-
cific heat at the order-disorder temperature of 6-brass may be explained by taking into account the vibrational entropy. Kirkwood
[2] have attempted,
brational entropy in
Some years later Wojtowicz and
in a more elaborated way,
to incorporate the vi-
a statistical mechanical treatment of the order-disorder
transformation of a binary alloy. However,
afterwards not much work has been
done on this subject,
mainly because of the great mathematical difficulties
which are involved in
an exact calculation.
For isolated effects,
it
has been possible by using a Green's function
formalism to calculate the change in the phonon spectrum and as a consequence the change in
the vibrational entropy
(see e.g.
[3]).
However,
this method is
not suitable when a large amount of "defects" (anti-site atoms)
is
generated,
for example when an ordered alloy disorders. Recently Bakker [4,51 has developed a new numerical technique to calculate "exactly" changes in
the vibrational entropy. This method is
especially
profitable for calculations on order-disorder systems. For the one-dimensional Mat.
Res. Soc. Symp.
Proc. Vol. 21 (1984) CElsevier Science Publishing Co.,
Inc.
344
chain and the two-dimensional square lattice results have been obtained using a model of central and non-central forces [41 and a model of first
and second
nearest-neighbour forces [5]. Now results will be given for a three-dimensional simple-cubic alloy. These results will be compared with the vibrational entropy change as calculated by the simpli
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